lecture 5 - eneagrid · ifts intensive course on advanced plasma physics-spring 2015, theory and...

IFTS Intensive Course on Advanced Plasma Physics-Spring 2015 Lecture 5 – 1

Lecture 5

Nonlinear beam-plasma dynamics with multiple resonances

Fulvio Zonca

http://www.afs.enea.it/zonca

ENEA C.R. Frascati, C.P. 65 - 00044 - Frascati, Italy.

Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou 310027, P.R.C.

May 13.th, 2015

IFTS Intensive Course on Advanced Plasma Physics-Spring 2015,Theory and simulation of nonlinear physics of the beam-plasma system

5–15 May 2015, IFTS – ZJU, Hangzhou

Fulvio Zonca


Chirikov criterion for resonance overlap

Chirikov was the first one providing a physical criterion for the onset ofchaotic motion in deterministic Hamiltonian systems (Chirikov 1959).

This criterion was applied to explain puzzling experimental results onplasma confinement in magnetic traps

K =

(

∆ωr

∆d

)2

> 1 .

Here, ∆ωr is the resonance width, ∝ δφ1/2 when computed in the isolatedresonance limit (pendulum approximation; see Lecture 3). Meanwhile, ∆d

is the frequency difference between unperturbed resonances.

For the beam-plasma system ∆ωr ≃ 3ωp(nB/2n)1/3 (see Lecture 3).

Fulvio Zonca

http://www.afs.enea.it/zonca/references/seminars/IFTS_spring15/lecture3.pdf



The multi-bleam plasma system

If multiple beams are characterized by ∆d > ∆ωr, the beams behave as inde-pendent and the nonlinear dynamics is equivalent to n-uncoupled oscillators(pendulums).

Fulvio Zonca


If ∆ωr > ∆d, resonances are coupled (Chirikov 1959). Since nonlinearsaturation of the simple beam-plasma system implies beam-heating (seeLecture 2), it is possible to assume that n beams are “cold” (Shapiro 1963).

The cold mutli-beam plasma system is a good paradigm for a warm-beamnonlinearly interacting with background plasma. However, one needs tostudy the system of n beams coupled with m > n self-consistent nonlin-ear oscillators, due to the generation of near-resonant sidebands (see later;Carlevaro 2015).

Fulvio Zonca



Vlasov and O’Neil formulations

The warm-beam plasma interaction can be simulated with the same Vlasov-Poisson equations adopted in Lecture 2 by simply accounting for a spectrumof Langmuir waves, ǫ(ωj, kj) = 0, with ωj ≃ ωp and j = 1, ..., m.

The approach based on n beams coupled with m > n self-consistent non-linear oscillators must be derived as extension of the original approach byO’Neil et al 1971 (Lecture 3).

Derivation follows Carlevaro 2015. Advantages:

• Spectrum of m Langmuir waves is not selected arbitrarily: the min-imal choice is of m = n plasma waves degenerated with the corre-sponding beam modes ωp = kjvDj , j = 1, ..., n

• Control on the nonlinear dynamics through spectral density (locationof resonances) and spectral intensity (saturation amplitudes of theoscillators in the uncoupled limit)

• Vlasov and multi-beam approach a la O’Neil coincide in the meanfield limit (Klimontovich, see Lecture 4)

Fulvio Zonca





Nonlinear evolution of n-resonant beams

Adopt a 1D model with the thermal plasma treated as linear dielectricmedium and n cold electron beams. Assume that the system is periodicwith finite size L.

The motion along the x direction is periodic, and is labeled by x1, x2, ...,xn for each beam, respectively.

Decomposing the single beams in N1, N2, ..., Nn charge sheets located at x1i,x2i, ..., xni, the system can be discretized using the following total chargedensity

ρB(x, t) = −e nB

(Lσ1

N1

N1∑

i=1

δ(x− x1i) + ...+Lσn

Nn

Nn∑

i=1

δ(x− xni))

,

where nB is the total suprathermal particle number density while σ1nB, ...,σnnB are the number densities of each beam.

Fulvio Zonca


The Langmuir wave scalar potential can be expressed using the Fourierrepresentation as

ϕ(x, t) =m∑

j=1

(

ϕj(kj, t)eikjx−iωpt + c.c.

)

,

where ϕj(kj, t) are assumed to be slowly varying fields and we have explicitlyindicated the time dependence relative to the rapid oscillations associatedto the plasma frequency.

The trajectories of charge sheets are derived from the equation of motionx = e∇ϕ/me as

x1i =ie

me

m∑

j=1

kjϕjeikjx1i−iωpt + c.c. ,

...

xni =ie

me

m∑

j=1

kjϕjeikjxni−iωpt + c.c. .

Fulvio Zonca


We now write Poisson’s equation in Fourier space, by projecting on the∼ eikjx plane wave

k2j ǫ(ωj, kj)ϕje

−iωpt = 4π ρBj(kj, t) ,

ρBj(kj, t) =1

L

∫ L

0

dx ρB(x, t)e−ikjx = −e nB

( σ1

N1

N1∑

i=1

e−ikjx1i+...+σn

Nn

Nn∑

i=1

e−ikjxni

)

.

Using the fact that m (nearly degenerate) Langmuir waves satisfy

ǫ(ωj, kj) ≃ ǫ(ωp, kj) +2

ωp(i∂t + ωp) ≃

2

ωp(i∂t + ωp) ,

the Poisson’s equation becomes

2

ωp

ϕj = i4πenB

k2j

( σ1

N1

N1∑

i=1

e−ikjx1i+iωpt + ...+σn

Nn

Nn∑

i=1

e−ikjx1i+iωpt)

.

Fulvio Zonca


Nonlinear equations in dimensionless form

Introduce the nonlinear displacement ζ of charged particle sheets, separatingthe unperturbed (free streaming) motion

x1i = v1t+ ζ1i(t) , x2i = v2t+ ζ2i(t) , ... , xni = vnt+ ζni(t) .

Introduce the scaled variables (see Lecture 2)

kj = 2πℓj/L , ζni = 2πζni/L , τ = tωpη , φj = [ϕj ek2j ]/[mη2ω2

p] ,

with η = (nB/2np)1/3 and the frequency mismatches (Note: βjj ≡ 0)

β1j = [kjv1 − ωp]/ωpη , ... , βnj = [kjvn − ωp]/ωpη .

E: Discuss the connection of these normalizations with those used in Lecture 3.What is the physical meaning of frequency mismatch? Why do we have βjj ≡ 0?

Fulvio Zonca




Final form of nonlinear dimensionless equations

ζ ′′1i = im∑

j=1

ℓ−1j φj e

iℓj ζ1i+iβ1jτ + c.c. ,

...

ζ ′′ni = im∑

j=1

ℓ−1j φj e

iℓj ζni+iβnjτ + c.c. ,

φ′

j = i( σ1

N1

N1∑

i=1

e−iℓj ζ1i−iβ1jτ + ...+σn

Nn

Nn∑

i=1

e−iℓj ζni−iβnjτ)

.

E: Derive these equations step by step.

Fulvio Zonca


Conservation laws and linear dispersion relation

Conservation laws are derived and discussed in Lecture 3 for the singlebeam case. Adopting the same approach, one can derive the momentumconservation

m∑

j=1

|φj|2

ℓj+

d

dτ

[

σ1

N1

N1∑

i=1

ζ1i + ...+σn

Nn

Nn∑

i=1

ζni

]

= 0 .

Meanwhile, energy conservation can be expressed as

m∑

j=1

|φj|2

ηℓ2j+ 2

m∑

j=1

ΩjRe|φj|

2

ℓ2j

+1

2

[

σ1

N1

N1∑

i=1

(

ζ ′1i +2π

L

v1ηωp

)2

+ ...+σn

Nn

Nn∑

i=1

(

ζ ′ni +2π

L

vnηωp

)2]

= 0

Fulvio Zonca



In these equations, we have used the definition (see Lecture 3)

φj(τ) = φj(0) exp

(

−i

∫ τ

0

Ωj(τ′)dτ ′

)

We have also made use of the identity

im∑

j=1

(

φ∗

jφ′′

j − φjφ′′∗

j

)

ℓ2j= 2

d

dτ

m∑

j=1

ΩjRe|φj|

2

ℓ2j

E: Derive these conservation laws step by step and show that they reduce to theconservation laws discussed in Lecture 3 for the case of a single beam.

E: Why do you need to invoke momentum conservation to recover the single beam“energy” conservation?

Fulvio Zonca




E: Recall that nonlinear displacements ζ are expressed in the (different) beammoving frames. Recall the normalization condition of the fields φj. Can youarticulate why the dependences on ℓj are those you should expect and why theyare consistent with the conservation laws of Lecture 2?

The linear dispersion relation is obtained noting φ′

j = −iΩjφj and that,denoting beams by the subscript a, by perturbation expansion

ζai = ζ0ai + ζ ′0ai τ − im∑

j=1

ℓ−1j φj exp[iℓj ζ

0ai + iℓj ζ

′0aiτ + iβajτ ]

(Ωj − βaj − ℓj ζ ′0ai)2

+ c.c. .

Choosing ζ ′0ai = 0 and noting∑

i eiℓj ζ

0

ai = 0 for uniformly distributed particles(see Hands on Session 3), the dispersion relation reduces to

Ωj =σ1

(Ωj − β1j)2+

σ2

(Ωj − β2j)2+ ...+

σn

(Ωj − βnj)2.

E: Derive the dispersion relation step by step.

Fulvio Zonca


http://www.afs.enea.it/zonca/references/seminars/IFTS_spring15/handson3.pptx


Numerical simulation results for two beams

Consider two beams with Θ ≡ v2/v1 < 1 and, by definition k2/k1 = ℓ2/ℓ1 =1/Θ > 1. Fixed parameter η = 0.01. Thus, natural beam resonance widthis (see p. 2) ∆ωr/ω ≃ ∆v/v ≃ 0.03σ1/3.

0 10 20 30 400.0

0.2

0.4

0.6

0.8

1.0

1.2

Τ

ÈΦ1ÈHs

olidLÈΦ

2ÈHd

ashe

dLQ=0.95 @1=1000 2=1031D Σ1=0.5 Σ2=0.5

Fulvio Zonca


Sufficiently separated beams behave independently, as expected.

0.000 0.005 0.010 0.015 0.020-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

Ξ

Ξ'

Q=0.95 Σ1=0.5 Σ2=0.5Τ=12.

Fulvio Zonca


Evidence of interacting beams. (i): faster beam mode is depleted

0 10 20 30 400.0

0.2

0.4

0.6

0.8

1.0

1.2

Τ

ÈΦ1ÈHs

olidLÈΦ

2ÈHd

ashe

dL

Q=0.97 @1=1000 2=1031D Σ1=0.5 Σ2=0.5

E: Explain why, in your opinion, the faster beam mode is depleted of energy, andnot the slower one.

Fulvio Zonca


Evidence of interacting beams. (ii)

0.000 0.005 0.010 0.015 0.020

-0.004

-0.002

0.000

0.002

0.004

Ξ

Ξ'

Q=0.98 Σ1=0.5 Σ2=0.5Τ=12.4

Fulvio Zonca


Evidence of interacting beams. (iii): σ1 > σ2; unbalanced beams

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

1.2

Τ

ÈΦ1ÈHs

olidLÈΦ

2ÈHd

ashe

dL

Q=0.96 @1=1000 2=1041D Σ1=0.8 Σ2=0.2

E: With σ1 > σ2, it is still the faster beam mode that is depleted of energy. Canyou explain why? Is the ratio of saturated amplitudes what you expect?

Fulvio Zonca


Back to using Lagrangian Coherent Structures

Use of LCS introduced in Hands on Session 1 and Lecture 4 and computedfrom FTLE.

σ(ξ, ξ′, τ,∆τ) = ln (δ∆τ/δτ )/∆τ .

When considering forward time evolution ∆τ > 0, the curves where theFTLE field is peaked define a repulsive transport barrier.

When considering backward time evolution ∆τ < 0, the curves where theFTLE field is peaked represent an attractive transport barrier.

Approximated 1D structures which correspond to the LCSs can be built byplotting the maximum values of σ(ξ, ξ′, τ,∆τ) as extracted from a contourplot of the reduced phase-space surface of section (see Lecture 1).

E: Comment about the sign of FTLE σ for forward and backward integration intime. Can you justify the definition of repulsive and attractive barriers?

Fulvio Zonca

http://www.afs.enea.it/zonca/references/seminars/IFTS_spring15/handson1.pptx




Evidence that LCS computed from FTLE describe transport barriers on afinite amount of time ∆t, as they evolve in time t.

This property can be used to determine the criterion for non-negligibleinteraction between two cold beams.

Fulvio Zonca


The value of Θ controls the beam merging: (i) independent beams

Q=0.95 Τ=11.6 Q=0.96 Τ=11.6

Fulvio Zonca


The value of Θ controls the beam merging: (ii) interacting beams

Q=0.97 Τ=11.6 Q=0.98 Τ=11.6

The criterion for interacting beams is consistent with |1 − Θ| ≃ ∆ωr/ω ≃∆v/v ≃ 0.03 (see p. 2 and p. 14).

Fulvio Zonca


Sideband generation in the beam-plasma system

Consider the system of two beams and two nearly degenerate Langmuirwaves, discussed so far.

In addition to k1 and k2 modes (k1 < k2; ∆k = k2 − k1 > 0), we havethe obvious beat wave generation at k−

3 = k1 − ∆k = 2k1 − k2 and k+3 =

k2 +∆k = 2k2 − k1.

Once the mechanism for driving nearest sidebands is identified, the pro-cess can repeat nonlinearly and generate a whole nonlinear spectrum ofmodes, stemming from the original two, resonantly driven by the two driv-ing suprathermal beams.

Efficient excitation of sideband modes is clearly not independent of theproperties of driving suprathermal particle source.

Investigate sideband generation mechanism for two beams assuming inde-pendent/interacting beams.

Fulvio Zonca


Inefficient sideband generation: (i) independent beams

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

1.2

Τ

Q=0.96 Σ1=0.5 Σ2=0.5

ÈΦ1ÈÈΦ2È

ÈΦ3+ÈÈΦ3

-È

Fulvio Zonca


Efficient sideband generation: (ii) interacting beams

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

1.2

Τ

Q=0.97 Σ1=0.5 Σ2=0.5

ÈΦ1È

ÈΦ2È

ÈΦ3+È

ÈΦ3-È

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

1.2

Τ

Q=0.98 Σ1=0.5 Σ2=0.5

ÈΦ1È

ÈΦ2È

ÈΦ3+È

ÈΦ3-È

The k1 mode is depleted of energy in amount depending on the initial beamvelocity difference. The amplitude of the k2 mode is weakly affected by thisprocess.

E: Explain these numerical results on the basis of your understanding of conser-vation laws discussed in this lecture as well as Lecture 2 and Lecture 3.

Fulvio Zonca




Efficient sideband generation: (iii) independent beams

0 10 20 30 400.0

0.2

0.4

0.6

0.8

1.0

1.2

Τ

Q=0.497 Σ1=0.5 Σ2=0.5

ÈΦ1È

ÈΦ2È

ÈΦbÈ

This special case occurs when Θ → 1/2−; i.e. k2 → 2k+1 and kb = k2−k1 →

k+1 . The “beat” kb mode is driven at low frequency, but it may be seed of

a near-resonant high frequency.

Fulvio Zonca


References and reading material

B. V. Chirikov, Resonance processes in magnetic traps, At. Energ. 6, 630 (1959)[Engl. Transl., J. Nucl. Energy Part C: Plasma Phys. 1, 253 (1960)].

B. V. Chirikov, A universal instability of many-dimensional oscillator systems,Phys. Rep. 52, 263 (1979).

V.D. Shapiro, Zh. Eksp. Teor. Fiz. 44 613 (1963) [Sov. Phys. JETP 17 416(1963)]

N. Carlevaro, M. V. Falessi, G. Montani and F. Zonca, Non-linear physics and

transport features of the beam-plasma instability, submitted to J. Plasma Phys.(2015).

Fulvio Zonca

lecture 5 - eneagrid · ifts intensive course on advanced plasma physics-spring 2015, theory and...

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